arXiv subjects

Alexander Katsevich

Publications and source records attributed to Alexander Katsevich.

32 records · Page 2Linked to original sources

arXiv2019

Diagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann-Hilbert approach

In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms $\mathcal{H}_L:L^2([b_L,0])\to L^2([0,b_R])$ and $\mathcal{H}_R:L^2([0,b_R])\to L^2([b_L,0])$. These operators arise when one studies the interior problem of tomography. The diagonalization of $\mathcal{H}_R,\mathcal{H}_L$ has been previously obtained, but only asymptotically when $b_L\neq-b_R$. We implement a novel approach based on the method of matrix Riemann-Hilbert problems (RHP) which diagonalizes $\mathcal{H}_R,\mathcal{H}_L$ explicitly. We also find the asymptotics of the solution to a related RHP and obtain error estimates.

math.CA↗

arXiv2019

Resolution analysis of inverting the generalized Radon transform from discrete data in $\mathbb R^3$

A number of practically important imaging problems involve inverting the generalized Radon transform (GRT) $\mathcal R$ of a function $f$ in $\mathbb R^3$. On the other hand, not much is known about the spatial resolution of the reconstruction from discretized data. In this paper we study how accurately and with what resolution the singularities of $f$ are reconstructed. The GRT integrates over a fairly general family of surfaces $\mathcal S_y$ in $\mathbb R^3$. Here $y$ is the parameter in the data space, which runs over an open set $\mathcal V\subset\mathbb R^3$. Assume that the data $g(y)=(\mathcal R f)(y)$ are known on a regular grid $y_j$ with step-sizes $O(ε)$ along each axis, and suppose $\mathcal S=\text{singsupp}(f)$ is a piecewise smooth surface. Let $f_ε$ denote the result of reconstruction from the descrete data. We obtain explicitly the leading singular behavior of $f_ε$ in an $O(ε)$-neighborhood of a generic point $x_0\in\mathcal S$, where $f$ has a jump discontinuity. We also prove that under some generic conditions on $\mathcal S$ (which include, e.g. a restriction on the order of tangency of $\mathcal S_y$ and $\mathcal S$), the singularities of $f$ do not lead to non-local artifacts. For both computations, a connection with the uniform distribution theory turns out to be important. Finally, we present a numerical experiment, which demonstrates a good match between the theoretically predicted behavior and actual reconstruction.

math.NA↗

arXiv2019

Analysis of reconstruction from discrete Radon transform data in $\mathbb R^3$ when the function has jump discontinuities

In this paper we study reconstruction of a function $f$ from its discrete Radon transform data in $\mathbb R^3$ when $f$ has jump discontinuities. Consider a conventional parametrization of the Radon data in terms of the affine and angular variables. The step-size along the affine variable is $ε$, and the density of measured directions on the unit sphere is $O(ε^2)$. Let $f_ε$ denote the result of reconstruction from the discrete data. Pick any generic point $x_0$ (i.e., satisfying some mild conditions), where $f$ has a jump. Our first result is an explicit leading term behavior of $f_ε$ in an $O(ε)$-neighborhood of $x_0$ as $ε\to0$. A closely related question is why can we accurately reconstruct functions with discontinuities at all? This is a fundamental question, which has not been studied in the literature in dimensions three and higher. We prove that the discrete inversion formula `works', i.e. if $x_0\not\in S:=\text{singsupp}(f)$ is generic, then $f_ε(x_0)\to f(x_0)$ as $ε\to0$. The proof of this result reveals a surprising connection with the theory of uniform distribution (u.d.). This is a new phenomenon that has not been known previously. We also present some numerical experiments, which confirm the validity of the developed theory.

math.NA↗

arXiv2018

Interior X-ray diffraction tomography with low-resolution exterior information

X-ray diffraction tomography (XDT) resolves spatially-variant XRD profiles within macroscopic objects, and provides improved material contrast compared to the conventional transmission-based computed tomography (CT). However, due to the small diffraction cross-section, XDT suffers from long imaging acquisition time, which could take tens of hours for a full scan using a table-top X-ray tube. In medical and industrial imaging applications, oftentimes only the XRD measurement within a region-of-interest (ROI) is required, which, together with the demand to reduce imaging time and radiation dose to the sample, motivates the development of interior XDT systems that scan and reconstruct only an internal region within the sample. The interior problem does not have a unique solution, and a direct inversion on the truncated projection data often leads to large reconstruction errors in ROI. To reduce the truncation artifacts, conventional attenuation-based interior reconstruction problems rely on a known region or piecewise constant constraint within the ROI, which do not apply to all the samples in XDT. Here we propose a quasi-interior XDT scheme that incorporates a small fraction of projection information from the exterior region to assist interior reconstruction. The low-resolution exterior projection data obviates the requirement for prior knowledge on the object, and allows the ROI reconstruction to be performed with the fast, widely-used filtered back-projection algorithm for easy integration into real-time XDT imaging modules.

physics.med-ph↗

arXiv2018

Inversion formula and range conditions for a vector multi-interval finite Hilbert transform in $L^2$

Given $n$ disjoint intervals $I_j$, on $\mathbb R$ together with $n$ functions $ψ_j\in L^2(I_j)$, $j=1,\dots n$, and an $n\times n$ matrix $Θ$, the problem is to find an $L^2$ solution $\vec φ= {\rm Col} (φ_1,\dots, φ_n)$, $φ_j \in L^2(I_j)$ to the linear system $χΘ\mathcal H \vec φ= \vecψ$, where $\mathcal H = {\rm diag} (\mathcal H_1 ,\dots, \mathcal H_n)$ is a matrix of finite Hilbert transforms and $χ=\text{diag}(χ_1,\dots,χ_n)$ is a matrix of the corresponding characteristic functions on $I_j$, and $\vec ψ={\rm Col} (ψ_1,\dots,ψ_n)$. Since we can interpret $χΘ\mathcal H \vec φ$ as a generalized vector multi-interval finite Hilbert transform, we call the formula for the solution as "the inversion formula" and the necessary and sufficient conditions for the existence of a solution as the "range conditions". In this paper we derive the explicit inversion formula and the range conditions in two specific cases: a) the matrix $Θ$ is symmetric and positive definite, and; b) all the entries of $Θ$ are equal to one. We also prove the uniqueness of solution, that is, that our transform is injective. When the matrix $Θ$ is positive definite, the inversion formula is given in terms of the solution of the associated matrix Riemann-Hilbert Problem. We also discuss other cases of the matrix $Θ$.

math.FA↗

arXiv2016

Reconstruction algorithms for a class of restricted ray transforms without added singularities

Let $X$ and $X^*$ denote a restricted ray transform along curves and a corresponding backprojection operator, respectively. Theoretical analysis of reconstruction from the data $Xf$ is usually based on a study of the composition $X^* D X$, where $D$ is some local operator (usually a derivative). If $X^*$ is chosen appropriately, then $X^* D X$ is a Fourier Integral Operator (FIO) with singular symbol. The singularity of the symbol leads to the appearance of artifacts (added singularities) that can be as strong as the original (or, useful) singularities. By choosing $D$ in a special way one can reduce the strength of added singularities, but it is impossible to get rid of them completely. In the paper we follow a similar approach, but make two changes. First, we replace $D$ with a nonlocal operator $\tilde D$ that integrates $Xf$ along a curve in the data space. The result $\tilde D Xf$ resembles the generalized Radon transform $R$ of $f$. The function $\tilde D Xf$ is defined on pairs $(x_0,Θ)\in U\times S^2$, where $U\subset\mathbb R^3$ is an open set containing the support of $f$, and $S^2$ is the unit sphere in $\mathbb R^3$. Second, we replace $X^*$ with a backprojection operator $R^*$ that integrates with respect to $Θ$ over $S^2$. It turns out that if $\tilde D$ and $R^*$ are appropriately selected, then the composition $R^* \tilde D X$ is an elliptic pseudodifferential operator of order zero with principal symbol 1. Thus, we obtain an approximate reconstruction formula that recovers all the singularities correctly and does not produce artifacts. The advantage of our approach is that by inserting $\tilde D$ we get access to the frequency variable $Θ$. In particular, we can incorporate suitable cut-offs in $R^*$ to eliminate bad directions $Θ$, which lead to added singularities.

math.CA↗

arXiv2016

An improved exact inversion formula for cone beam vector tomography

In this article we present an improved exact inversion formula for the 3D cone beam transform of vector fields. It is well known that only the solenoidal part of a vector field can be determined by the longitudinal ray transform of a vector field in cone beam geometry. The exact inversion formula, as it was developed in A. Katsevich and T. Schuster, 'An exact inversion formula for cone beam vector tomography', Inverse Problems 29 (2013), consists of two parts. the first part is of filtered backprojection type, whereas the second part is a costly 4D integration and very inefficient. In this article we tackle this second term and achieve an improvement which is easily to implement and saves one order of integration. The theory says that the first part contains all information about the curl of the field, whereas the second part presumably has information about the boundary values. This suggestion is supported by the fact that the second part vanishes if the exact field is divergence free and tangential at the boundary. A number of numerical tests, that are also subject of this article, confirm the theoretical results and the exactness of the formula.

math.NA↗

arXiv2016

Stability estimates for the regularized inversion of the truncated Hilbert transform

In limited data computerized tomography, the 2D or 3D problem can be reduced to a family of 1D problems using the differentiated backprojection (DBP) method. Each 1D problem consists of recovering a compactly supported function $f \in L^2(\mathcal F)$, where $\mathcal F$ is a finite interval, from its partial Hilbert transform data. When the Hilbert transform is measured on a finite interval $\mathcal G$ that only overlaps but does not cover $\mathcal F$ this inversion problem is known to be severely ill-posed [1]. In this paper, we study the reconstruction of $f$ restricted to the overlap region $\mathcal F \cap \mathcal G$. We show that with this restriction and by assuming prior knowledge on the $L^2$ norm or on the variation of $f$, better stability with Hölder continuity (typical for mildly ill-posed problems) can be obtained.

math.CA↗

arXiv2015

Diagonalization of the Finite Hilbert Transform on two adjacent intervals

We study the interior problem of tomography. The starting point is the Gelfand-Graev formula, which converts the tomographic data into the finite Hilbert transform (FHT) of an unknown function $f$ along a collection of lines. Pick one such line, call it the $x$-axis, and assume that the function to be reconstructed depends on a one-dimensional argument by restricting $f$ to the line. Let $Ω_1$ be the interval where $f$ is supported, and $Ω_2$ be the interval where the Hilbert transform of $f$ can be computed using the Gelfand-Graev formula. The equation we study is $H_1 f=g|_{Ω_2}$, where $H_1$ is the FHT that integrates over $Ω_1$ and gives the result on $Ω_2$, i.e. $H_1: L^2(Ω_1)\to L^2(Ω_2)$. In the case of the interior problem the tomographic data are truncated, and $Ω_1$ is no longer a subset of $Ω_2$. In this paper we consider the case when the intervals $Ω_1=(a_1,0)$ and $Ω_2=(0,a_2)$ are adjacent. Here $a_1 < 0 < a_2$. First we find a differential operator $L$ that commutes with $H_1$. Using the Titchmarsh-Weyl theory, we show that $L$ has only continuous spectrum and obtain two isometric transformations $U_1$, $U_2$, such that $U_2 H_1 U_1^*$ is the multiplication operator with the function $σ(λ)$, $λ\geq(a_1^2+a_2^2)/8$. Here $λ$ is the spectral parameter. Then we show that $σ(λ)\to0$ as $λ\to\infty$ exponentially fast. We also obtain the leading asymptotic behavior of the kernels involved in the integral operators $U_1$, $U_2$ as $λ\to\infty$. When the intervals are symmetric, i.e. $-a_1=a_2$, the operators $U_1$, $U_2$ are obtained explicitly in terms of hypergeometric functions.

math.CA↗

arXiv2014

Covariance Matrix Estimation for the Cryo-EM Heterogeneity Problem

In cryo-electron microscopy (cryo-EM), a microscope generates a top view of a sample of randomly-oriented copies of a molecule. The problem of single particle reconstruction (SPR) from cryo-EM is to use the resulting set of noisy 2D projection images taken at unknown directions to reconstruct the 3D structure of the molecule. In some situations, the molecule under examination exhibits structural variability, which poses a fundamental challenge in SPR. The heterogeneity problem is the task of mapping the space of conformational states of a molecule. It has been previously suggested that the leading eigenvectors of the covariance matrix of the 3D molecules can be used to solve the heterogeneity problem. Estimating the covariance matrix is challenging, since only projections of the molecules are observed, but not the molecules themselves. In this paper, we formulate a general problem of covariance estimation from noisy projections of samples. This problem has intimate connections with matrix completion problems and high-dimensional principal component analysis. We propose an estimator and prove its consistency. When there are finitely many heterogeneity classes, the spectrum of the estimated covariance matrix reveals the number of classes. The estimator can be found as the solution to a certain linear system. In the cryo-EM case, the linear operator to be inverted, which we term the projection covariance transform, is an important object in covariance estimation for tomographic problems involving structural variation. Inverting it involves applying a filter akin to the ramp filter in tomography. We design a basis in which this linear operator is sparse and thus can be tractably inverted despite its large size. We demonstrate via numerical experiments on synthetic datasets the robustness of our algorithm to high levels of noise.

math.NA↗

arXiv2014

Singular value decomposition of a finite Hilbert transform defined on several intervals and the interior problem of tomography: the Riemann-Hilbert problem approach

We study the asymptotics of singular values and singular functions of a Finite Hilbert transform (FHT), which is defined on several intervals. Transforms of this kind arise in the study of the interior problem of tomography. We suggest a novel approach based on the technique of the matrix Riemann-Hilbert problem and the steepest descent method of Deift-Zhou. We obtain a family of matrix RHPs depending on the spectral parameter $λ$ and show that the singular values of the FHT coincide with the values of $λ$ for which the RHP is not solvable. Expressing the leading order solution as $λ\to 0$ of the RHP in terms of the Riemann Theta functions, we prove that the asymptotics of the singular values can be obtained by studying the intersections of the locus of zeroes of a certain Theta function with a straight line. This line can be calculated explicitly, and it depends on the geometry of the intervals that define the FHT. The leading order asymptotics of the singular functions and singular values are explicitly expressed in terms of the Riemann Theta functions and of the period matrix of the corresponding normalized differentials, respectively. We also obtain the error estimates for our asymptotic results.

math.SP↗

arXiv2014

On Sobolev instability of the interior problem of tomography

In this paper we continue investigation of the interior problem of tomography that was started in \cite{BKT2}. As is known, solving the interior problem {with prior data specified on a finite collection of intervals $I_i$} is equivalent to analytic continuation of a function from $I_i$ to an open set ${\bf J}$. In the paper we prove that this analytic continuation can be obtained with the help of a simple explicit formula, which involves summation of a series. Our second result is that the operator of analytic continuation is not stable for any pair of Sobolev spaces regardless of how close the set ${\bf J}$ is to $I_i$. Our main tool is the singular value decomposition of the operator $\mathcal H^{-1}_e$ that arises when the interior problem is reduced to a problem of inverting the Hilbert transform from incomplete data. The asymptotics of the singular values and singular functions of $\mathcal H^{-1}_e$, the latter being valid uniformly on compact subsets {of the interior of $I_i$}, was obtained in \cite{BKT2}. {Using these asymptotics we can accurately measure the degree of ill-posedness of the analytic continuation as a function of the target interval ${\bf J}$.} Our {last} result is the convergence of the asymptotic approximation of the singular functions {in the $L^2(I_i)$ sense}.

math-ph↗

arXiv2013

Asymptotic analysis of the SVD for the truncated Hilbert transform with overlap

The truncated Hilbert transform with overlap $H_T$ is an operator that arises in tomographic reconstruction from limited data, more precisely in the method of Differentiated Back-Projection (DBP). Recent work [1] has shown that the singular values of this operator accumulate at both zero and one. To better understand the properties of the operator and, in particular, the ill-posedness of the inverse problem associated with it, it is of interest to know the rates at which the singular values approach zero and one. In this paper, we exploit the property that $H_T$ commutes with a second-order differential operator $L_S$ and the global asymptotic behavior of its eigenfunctions to find the asymptotics of the singular values and singular functions of $H_T$.

math.CA↗

arXiv2013

Spectral analysis of the truncated Hilbert transform with overlap

We study a restriction of the Hilbert transform as an operator $H_T$ from $L^2(a_2,a_4)$ to $L^2(a_1,a_3)$ for real numbers $a_1 < a_2 < a_3 < a_4$. The operator $H_T$ arises in tomographic reconstruction from limited data, more precisely in the method of differentiated back-projection (DBP). There, the reconstruction requires recovering a family of one-dimensional functions $f$ supported on compact intervals $[a_2,a_4]$ from its Hilbert transform measured on intervals $[a_1,a_3]$ that might only overlap, but not cover $[a_2,a_4]$. We show that the inversion of $H_T$ is ill-posed, which is why we investigate the spectral properties of $H_T$. We relate the operator $H_T$ to a self-adjoint two-interval Sturm-Liouville problem, for which we prove that the spectrum is discrete. The Sturm-Liouville operator is found to commute with $H_T$, which then implies that the spectrum of $H_T^* H_T$ is discrete. Furthermore, we express the singular value decomposition of $H_T$ in terms of the solutions to the Sturm-Liouville problem. The singular values of $H_T$ accumulate at both $0$ and $1$, implying that $H_T$ is not a compact operator. We conclude by illustrating the properties obtained for $H_T$ numerically.

math.FA↗